What is meant by mentioning "the Qur’an," "the Mushaf," and "the Book" in the hadiths and verses, if the Qur’an had not been compiled into a single Mushaf during the time of the Messenger (peace be upon him)?

The Quran was written during the time of the Prophet, peace and blessings be upon him, but it was scattered and not collected in a single mushaf (codex). Abu Bakr Al-Siddiq, may Allah be pleased with him, collected it into a single volume after the death of the Prophet, peace and blessings be upon him. Then, during the caliphate of Uthman, may Allah be pleased with him, copies were made into multiple masahif (codices). Any narrations that might suggest the Quran was collected in a single mushaf during the Prophet's time should be understood to mean scattered sheets on which parts of the Quran were written, or that they were prophetic statements about what would come to be. The designation of the Quran as a "book" does not imply that it was compiled into a single mushaf during the Prophet's time; rather, it refers to its comprehensive nature regarding stories and rulings, and Allah's emphasis on its writing and collection.